Optimal control of impulsive hybrid systems

Abstract

In this paper, we deal with optimization techniques for a class of hybrid systems that comprise continuous controllable dynamics and impulses (jumps) in the state. Using the mathematical techniques of distributional derivatives and impulse differential equations, we rewrite the original hybrid control system as a system with autonomous location transitions. For the obtained auxiliary dynamical system and the corresponding optimal control problem (OCP), we apply the Lagrange approach and derive the reduced gradient formulas. Moreover, we formulate necessary optimality conditions for the above hybrid OCPs, and discuss the newly elaborated Pontryagin-type Maximum Principle for impulsive OCPs. As in the case of the conventional OCPs, the proposed first order optimization techniques provide a basis for constructive computational algorithms.

Introduction

Hybrid systems are mathematical models of heterogeneous control systems consisting of a continuous part, a finite number of continuous controllers, and a discrete supervisor. These models can represent an extremely wide range of systems of practical interest (see e.g., [7]) and are accepted as realistic models, for instance, in industrial electronics, power systems engineering, maneuvering aircrafts, automotive control systems, controllable chemical processes and communication networks. The emergence of a hybrid systems modeling framework is providing a new perspective for some important modern control processes. Hybrid systems have been extensively studied in the past decade, both in theory and practice [1], [2], [3], [4], [5], [7], [8], [10], [12], [15], [16], [19], [20]. In this paper, we investigate a specific family of hybrid systems, and the corresponding hybrid OCPs. We refer to [1], [2], [3], [4], [5], [6], [8], [9], [10], [12], [19], [20], [22] for some alternative types of hybrid systems and optimization problems. The class of problems to be discussed in this work concerns hybrid systems, where discrete transitions are being triggered by the continuous dynamics, and are accompanied by discontinuous changes in the continuous state variable (state jumps) [1], [4], [16]. The control objective is to minimize a cost functional, where the control parameters are not only usual control inputs but also the (bounded) magnitudes of the above-mentioned state jumps.

A mathematically correct description of continuous-type impulsive dynamic systems, as a rule, incorporates impulse differential equations. In this paper, we study hybrid control systems governed by ordinary differential equations with distributional (week) derivatives, and consider these differential equations in the space of generalized functions (distributions). Using a simple coordinates transformation proposed in [11], [18], we rewrite the original impulsive hybrid control systems in the form of an auxiliary hybrid system with autonomous location transitions (see, e.g., [2], [3], [19]), and also replace the sophisticated original OCP by an auxiliary optimization problem. The type of auxiliary problems to be studied also concerns hybrid systems with externally forced switchings and measurable control input signals. On the other hand, the obtained auxiliary hybrid OCP is a jumps-free optimization problem. Using operator theoretic techniques and the Lagrange method, a general gradient formula for the auxiliary OCP can be derived. We discuss some necessary conditions of optimality in the form of the Pontryagin-type Maximum Principle for the original impulsive hybrid system, and cast the auxiliary OCP as a nonlinear programming problem for which gradient formulas are derived. This leads to the solution of more tractable problems, and gradient-descent methods can be used for constructing a desired control profile. The invertibility of the above-mentioned coordinate transformation makes it possible to apply the proposed gradient-based computational techniques to the original OCP governed by an impulsive hybrid system.

The outline of the paper is as follows. In Section 2, we introduce the main concepts, and formally describe the impulsive hybrid systems and the hybrid OCPs investigated in this contribution. Section 3 contains an equivalent representation of hybrid systems under consideration, and includes the auxiliary optimization problem for the given impulsive hybrid OCP. In Section 4, we present the Lagrange formalism for the auxiliary hybrid OCP and give the representation of the corresponding functional derivatives. In Section 5, we apply the obtained theoretic results and consider some related first order optimization techniques. Moreover, we also discuss the related Pontryagin-type Maximum Principle from [4]. Section 6 summarizes the paper.